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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Collinearity</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Colinear" redirects here. For the use in genetics, see <a href="Synteny" title="Synteny">synteny</a>. For the use in coalgebra theory, see <a href="Colinear_map" class="mw-redirect" title="Colinear map">colinear map</a>. For colinearity in statistics, see <a href="Multicollinearity" title="Multicollinearity">multicollinearity</a>.</div>

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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/collinearity" class="extiw external" title="wiktionary:collinearity">collinearity</a></b></i> or <i><b><a href="https://en.wiktionary.org/wiki/collinear" class="extiw external" title="wiktionary:collinear">collinear</a></b></i> in Wiktionary, the free dictionary.</div></div>

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<p>In <a href="Geometry" title="Geometry">geometry</a>, <b>collinearity</b> of a set of <a href="Point_(geometry)" title="Point (geometry)">points</a> is the property of their lying on a single <a href="Line_(geometry)" title="Line (geometry)">line</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A set of points with this property is said to be <b>collinear</b> (sometimes spelled as <b>colinear</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".

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<div class="mw-heading mw-heading2"><h2 id="Points_on_a_line">Points on a line</h2></div>

<p>In any geometry, the set of points on a line are said to be <b>collinear</b>. In <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a <a href="Line_(geometry)" title="Line (geometry)">line</a> is typically a <a href="Primitive_notion" title="Primitive notion">primitive (undefined) object type</a>, so such visualizations will not necessarily be appropriate. A <a href="Mathematical_model" title="Mathematical model">model</a> for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in <a href="Spherical_geometry" title="Spherical geometry">spherical geometry</a>, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being <i>in a row</i>.

</p><p>A mapping of a geometry to itself which sends lines to lines is called a <i><a href="Collineation" title="Collineation">collineation</a></i>; it preserves the collinearity property.

The <a href="Linear_map" title="Linear map">linear maps (or linear functions)</a> of <a href="Vector_spaces" class="mw-redirect" title="Vector spaces">vector spaces</a>, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In <a href="Projective_geometry" title="Projective geometry">projective geometry</a> these linear mappings are called <i><a href="Homography" title="Homography">homographies</a></i> and are just one type of collineation.

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<div class="mw-heading mw-heading2"><h2 id="Examples_in_Euclidean_geometry">Examples in Euclidean geometry</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Triangles">Triangles</h3></div>

<p>In any triangle the following sets of points are collinear:

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<ul><li>The <a href="Orthocenter" title="Orthocenter">orthocenter</a>, the <a href="Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a>, the <a href="Centroid" title="Centroid">centroid</a>, the <a href="Exeter_point" title="Exeter point">Exeter point</a>, the <a href="De_Longchamps_point" title="De Longchamps point">de Longchamps point</a>, and the center of the <a href="Nine-point_circle" title="Nine-point circle">nine-point circle</a> are collinear, all falling on a line called the <a href="Euler_line" title="Euler line">Euler line</a>.</li>

<li>The de Longchamps point also has <a href="De_Longchamps_point#Additional_properties" title="De Longchamps point">other collinearities</a>.</li>

<li>Any vertex, the tangency of the opposite side with an <a href="Excircle" class="mw-redirect" title="Excircle">excircle</a>, and the <a href="Nagel_point" title="Nagel point">Nagel point</a> are collinear in a line called a <a href="Splitter_(geometry)" title="Splitter (geometry)">splitter</a> of the triangle.</li>

<li>The midpoint of any side, the point that is equidistant from it along the triangle's boundary in either direction (so these two points <a href="Bisection#Area_bisectors_and_perimeter_bisectors" title="Bisection">bisect the perimeter</a>), and the <a href="Spieker_center" title="Spieker center">center of the Spieker circle</a> are collinear in a line called a <a href="Cleaver_(geometry)" title="Cleaver (geometry)">cleaver</a> of the triangle. (The <a href="Spieker_circle" title="Spieker circle">Spieker circle</a> is the <a href="Incircle" class="mw-redirect" title="Incircle">incircle</a> of the <a href="Medial_triangle" title="Medial triangle">medial triangle</a>, and <a href="Spieker_center" title="Spieker center">its center</a> is the <a href="Center_of_mass" title="Center of mass">center of mass</a> of the <a href="Perimeter" title="Perimeter">perimeter</a> of the triangle.)</li>

<li>Any vertex, the tangency of the opposite side with the incircle, and the <a href="Gergonne_point" class="mw-redirect" title="Gergonne point">Gergonne point</a> are collinear.</li>

<li>From any point on the <a href="Circumcircle" title="Circumcircle">circumcircle</a> of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the <a href="Simson_line" title="Simson line">Simson line</a> of the point on the circumcircle.</li>

<li>The lines connecting the feet of the <a href="Altitude_(triangle)" title="Altitude (triangle)">altitudes</a> intersect the opposite sides at collinear points.<sup id="cite_ref-Johnson_3-0" class="reference"><a href="#cite_note-Johnson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.199">: p.199 </span></sup></li>

<li>A triangle's <a href="Incenter" title="Incenter">incenter</a>, the midpoint of an <a href="Altitude_(geometry)" class="mw-redirect" title="Altitude (geometry)">altitude</a>, and the point of contact of the corresponding side with the <a href="Excircle" class="mw-redirect" title="Excircle">excircle</a> relative to that side are collinear.<sup id="cite_ref-AC_4-0" class="reference"><a href="#cite_note-AC-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.120, #78">: p.120, #78 </span></sup></li>

<li><a href="Menelaus'_theorem" class="mw-redirect" title="Menelaus' theorem">Menelaus' theorem</a> states that three points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1},P_{2},P_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{1},P_{2},P_{3}}</annotation>

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</math></span><img src="./da44480c4ce88a564e8ff52138d90061fbf91394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.707ex; height:2.509ex;" alt="{\displaystyle P_{1},P_{2},P_{3}}" loading="lazy"></span> on the sides (some <a href="Extended_side" title="Extended side">extended</a>) of a triangle opposite vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},A_{2},A_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},A_{3}}</annotation>

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</math></span><img src="./18eb93c88961a573677555bffd4d27cc0557e70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.46ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},A_{3}}" loading="lazy"></span> respectively are collinear if and only if the following products of segment lengths are equal:<sup id="cite_ref-Johnson_3-1" class="reference"><a href="#cite_note-Johnson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 147">: p. 147 </span></sup></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle P_{1}A_{2}\cdot P_{2}A_{3}\cdot P_{3}A_{1}=P_{1}A_{3}\cdot P_{2}A_{1}\cdot P_{3}A_{2}.}</annotation>

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</math></span><img src="./2640e614329b8a525803fa3917f871b4fac1f288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.525ex; height:2.509ex;" alt="{\displaystyle P_{1}A_{2}\cdot P_{2}A_{3}\cdot P_{3}A_{1}=P_{1}A_{3}\cdot P_{2}A_{1}\cdot P_{3}A_{2}.}" loading="lazy"></span></dd></dl></dd></dl>

<ul><li>The incenter, the centroid, and the Spieker circle's center are collinear.</li>

<li>The circumcenter, the <a href="Brocard_points#The_segment_between_the_first_two_Brocard_points" title="Brocard points">Brocard midpoint</a>, and the <a href="Lemoine_point" title="Lemoine point">Lemoine point</a> of a triangle are collinear.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li>

<li>Two <a href="Perpendicular_lines" class="mw-redirect" title="Perpendicular lines">perpendicular lines</a> intersecting at the <a href="Orthocenter" title="Orthocenter">orthocenter</a> of a triangle each intersect each of the triangle's <a href="Extended_side" title="Extended side">extended sides</a>. The midpoints on the three sides of these points of intersection are collinear in the <a href="Droz-Farny_line_theorem" title="Droz-Farny line theorem">Droz–Farny line</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Quadrilaterals">Quadrilaterals</h3></div>

<ul><li>In a convex <a href="Quadrilateral" title="Quadrilateral">quadrilateral</a> <span class="texhtml mvar" style="font-style:italic;">ABCD</span> whose opposite sides intersect at <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">F</span>, the <a href="Midpoint" title="Midpoint">midpoints</a> of <span class="texhtml mvar" style="font-style:italic;"><span style="text-decoration:overline;">AC</span>, <span style="text-decoration:overline;">BD</span>, <span style="text-decoration:overline;">EF</span></span> are collinear and the line through them is called the <a href="Newton_line" title="Newton line">Newton line</a>. If the quadrilateral is a <a href="Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a>, then its incenter also lies on this line.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>

<li>In a convex quadrilateral, the quasiorthocenter <span class="texhtml mvar" style="font-style:italic;">H</span>, the "area centroid" <span class="texhtml mvar" style="font-style:italic;">G</span>, and the quasicircumcenter <span class="texhtml mvar" style="font-style:italic;">O</span> are collinear in this order, and <span class="texhtml"><span style="text-decoration:overline;"><i>HG</i></span> = 2<span style="text-decoration:overline;"><i>GO</i></span></span>.<sup id="cite_ref-Myakishev_7-0" class="reference"><a href="#cite_note-Myakishev-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> (See <a href="Quadrilateral#Remarkable_points_and_lines_in_a_convex_quadrilateral" title="Quadrilateral">Quadrilateral#Remarkable points and lines in a convex quadrilateral</a>.)</li></ul>

<ul><li>Other collinearities of a <a href="Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a> are given in <a href="Tangential_quadrilateral#Collinear_points" title="Tangential quadrilateral">Tangential quadrilateral#Collinear points</a>.</li>

<li>In a <a href="Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>, the <a href="Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a>, the <a href="Quadrilateral#Remarkable_points_and_lines_in_a_convex_quadrilateral" title="Quadrilateral">vertex centroid</a> (the intersection of the two bimedians), and the <a href="Cyclic_quadrilateral#Anticenter_and_collinearities" title="Cyclic quadrilateral">anticenter</a> are collinear.<sup id="cite_ref-Honsberger_8-0" class="reference"><a href="#cite_note-Honsberger-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li>

<li>In a cyclic quadrilateral, the <a href="Quadrilateral#Remarkable_points_and_lines_in_a_convex_quadrilateral" title="Quadrilateral">area centroid</a>, the vertex centroid, and the intersection of the diagonals are collinear.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li>

<li>In a <a href="Tangential_trapezoid" title="Tangential trapezoid">tangential trapezoid</a>, the tangencies of the <a href="Incircle" class="mw-redirect" title="Incircle">incircle</a> with the two bases are collinear with the incenter.</li>

<li>In a tangential trapezoid, the midpoints of the legs are collinear with the incenter.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Hexagons">Hexagons</h3></div>

<ul><li><a href="Pascal's_theorem" title="Pascal's theorem">Pascal's theorem</a> (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a <a href="Conic_section" title="Conic section">conic section</a> (i.e., <a href="Ellipse" title="Ellipse">ellipse</a>, <a href="Parabola" title="Parabola">parabola</a> or <a href="Hyperbola" title="Hyperbola">hyperbola</a>) and joined by line segments in any order to form a <a href="Hexagon" title="Hexagon">hexagon</a>, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon. The converse is also true: the <a href="Braikenridge%E2%80%93Maclaurin_theorem" title="Braikenridge–Maclaurin theorem">Braikenridge–Maclaurin theorem</a> states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic, which may be degenerate as in <a href="Pappus's_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's hexagon theorem</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Conic_sections">Conic sections</h3></div>

<ul><li>By <a href="Monge's_theorem" title="Monge's theorem">Monge's theorem</a>, for any three <a href="Circle" title="Circle">circles</a> in a plane, none of which is completely inside one of the others, the three intersection points of the three pairs of lines, each externally tangent to two of the circles, are collinear.</li>

<li>In an <a href="Ellipse" title="Ellipse">ellipse</a>, the center, the two <a href="Focus_(geometry)" title="Focus (geometry)">foci</a>, and the two <a href="Vertex_(curve)" title="Vertex (curve)">vertices</a> with the smallest <a href="Radius_of_curvature_(mathematics)" class="mw-redirect" title="Radius of curvature (mathematics)">radius of curvature</a> are collinear, and the center and the two vertices with the greatest radius of curvature are collinear.</li>

<li>In a <a href="Hyperbola" title="Hyperbola">hyperbola</a>, the center, the two foci, and the two vertices are collinear.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Cones">Cones</h3></div>

<ul><li>The <a href="Center_of_mass" title="Center of mass">center of mass</a> of a <a href="Cone" title="Cone">conic solid</a> of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Tetrahedrons">Tetrahedrons</h3></div>

<ul><li>The centroid of a tetrahedron is the midpoint between its <a href="Tetrahedron#Properties_analogous_to_those_of_a_triangle" title="Tetrahedron">Monge point</a> and <a href="Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a>. These points define the <i>Euler line</i> of the tetrahedron that is analogous to the <a href="Euler_line" title="Euler line">Euler line</a> of a triangle. The center of the <a href="Tetrahedron#Properties_analogous_to_those_of_a_triangle" title="Tetrahedron">tetrahedron's twelve-point sphere</a> also lies on the Euler line.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Algebra">Algebra</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Collinearity_of_points_whose_coordinates_are_given">Collinearity of points whose coordinates are given</h3></div>

<p>In <a href="Coordinate_geometry" class="mw-redirect" title="Coordinate geometry">coordinate geometry</a>, in <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of <a href="Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> 1 or less. For example, given three points

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X&=(x_{1},\ x_{2},\ \dots ,\ x_{n}),\\Y&=(y_{1},\ y_{2},\ \dots ,\ y_{n}),\\Z&=(z_{1},\ z_{2},\ \dots ,\ z_{n}),\end{aligned}}}</annotation>

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</math></span><img src="./dcfbc8d94fc1cddda22cd2ea24dbf441c29f8a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.943ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}X&=(x_{1},\ x_{2},\ \dots ,\ x_{n}),\\Y&=(y_{1},\ y_{2},\ \dots ,\ y_{n}),\\Z&=(z_{1},\ z_{2},\ \dots ,\ z_{n}),\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>if the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\\y_{1}&y_{2}&\dots &y_{n}\\z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}</annotation>

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</math></span><img src="./065562817eda799f6fb25314035d76f8e8671101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:20.859ex; height:9.509ex;" alt="{\displaystyle {\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\\y_{1}&y_{2}&\dots &y_{n}\\z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>is of <a href="Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> 1 or less, the points are collinear.

</p><p>Equivalently, for every subset of <span class="texhtml mvar" style="font-style:italic;">X, Y, Z</span>, if the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\dots &x_{n}\\1&y_{1}&y_{2}&\dots &y_{n}\\1&z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}</annotation>

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</math></span><img src="./0051b55c8e7c90552d9a551b3fe8d4b553f92160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.344ex; height:9.509ex;" alt="{\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\dots &x_{n}\\1&y_{1}&y_{2}&\dots &y_{n}\\1&z_{1}&z_{2}&\dots &z_{n}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>is of <a href="Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> 2 or less, the points are collinear. In particular, for three points in the plane (<span class="texhtml"><i>n</i> = 2</span>), the above matrix is square and the points are collinear if and only if its <a href="Determinant" title="Determinant">determinant</a> is zero; since that 3 × 3 determinant is plus or minus twice the <a href="Triangle#Using_coordinates" title="Triangle">area of a triangle</a> with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only if the triangle with those points as vertices has zero area.

</p>

<div class="mw-heading mw-heading3"><h3 id="Collinearity_of_points_whose_pairwise_distances_are_given">Collinearity of points whose pairwise distances are given</h3></div>

<p>A set of at least three distinct points is called <a href="Distance_geometry#Cayley–Menger_determinants" title="Distance geometry">straight</a>, meaning all the points are collinear, if and only if, for every three of those points <span class="texhtml mvar" style="font-style:italic;">A, B, C</span>, the following determinant of a <a href="Cayley%E2%80%93Menger_determinant" title="Cayley–Menger determinant">Cayley–Menger determinant</a> is zero (with <span class="texhtml"><i>d</i>(<i>AB</i>)</span> meaning the distance between <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, etc.):

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<annotation encoding="application/x-tex">{\displaystyle \det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&1\\d(AB)^{2}&0&d(BC)^{2}&1\\d(AC)^{2}&d(BC)^{2}&0&1\\1&1&1&0\end{bmatrix}}=0.}</annotation>

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</math></span><img src="./89c4138887595a95654a8015e8b1aa25d140e4ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:43.315ex; height:13.176ex;" alt="{\displaystyle \det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&1\\d(AB)^{2}&0&d(BC)^{2}&1\\d(AC)^{2}&d(BC)^{2}&0&1\\1&1&1&0\end{bmatrix}}=0.}" loading="lazy"></span></dd></dl></dd></dl>

<p>This determinant is, by <a href="Heron's_formula" title="Heron's formula">Heron's formula</a>, equal to −16 times the square of the area of a triangle with side lengths <span class="texhtml"><i>d</i>(<i>AB</i>), <i>d</i>(<i>BC</i>), <i>d</i>(<i>AC</i>)</span>; so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices <span class="texhtml mvar" style="font-style:italic;">A, B, C</span> has zero area (so the vertices are collinear).

</p><p>Equivalently, a set of at least three distinct points are collinear if and only if, for every three of those points <span class="texhtml mvar" style="font-style:italic;">A, B, C</span> with <span class="texhtml"><i>d</i>(<i>AC</i>)</span> greater than or equal to each of <span class="texhtml"><i>d</i>(<i>AB</i>)</span> and <span class="texhtml"><i>d</i>(<i>BC</i>)</span>, the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a> <span class="texhtml"><i>d</i>(<i>AC</i>) ≤ <i>d</i>(<i>AB</i>) + <i>d</i>(<i>BC</i>)</span> holds with equality.

</p>

<div class="mw-heading mw-heading2"><h2 id="Number_theory">Number theory</h2></div>

<p>Two numbers <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are not <a href="Coprime_numbers" class="mw-redirect" title="Coprime numbers">coprime</a>—that is, they share a common factor other than 1—if and only if for a rectangle plotted on a <a href="Square_lattice" title="Square lattice">square lattice</a> with vertices at <span class="texhtml">(0, 0), (<i>m</i>, 0), (<i>m</i>, <i>n</i>), (0, <i>n</i>)</span>, at least one interior point is collinear with <span class="texhtml">(0, 0)</span> and <span class="texhtml">(<i>m, n</i>)</span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Concurrency_(plane_dual)">Concurrency (plane dual)</h2></div>

<p>In various <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane geometries</a> the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called <a href="Duality_(projective_geometry)" title="Duality (projective geometry)">plane duality</a>. Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called <b>concurrency</b>, and the lines are said to be <a href="Concurrent_lines" title="Concurrent lines">concurrent lines</a>. Thus, concurrency is the plane dual notion to collinearity.

</p>

<div class="mw-heading mw-heading2"><h2 id="Collinearity_graph">Collinearity graph</h2></div>

<p>Given a <a href="Partial_geometry" title="Partial geometry">partial geometry</a> <span class="texhtml mvar" style="font-style:italic;">P</span>, where two points determine at most one line, a <b>collinearity graph</b> of <span class="texhtml mvar" style="font-style:italic;">P</span> is a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> whose vertices are the points of <span class="texhtml mvar" style="font-style:italic;">P</span>, where two vertices are <a href="Adjacent_vertex" class="mw-redirect" title="Adjacent vertex">adjacent</a> if and only if they determine a line in <span class="texhtml mvar" style="font-style:italic;">P</span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Usage_in_statistics_and_econometrics">Usage in statistics and econometrics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multicollinearity" title="Multicollinearity">Multicollinearity</a></div>

<p>In <a href="Statistics" title="Statistics">statistics</a>, <b>collinearity</b> refers to a linear relationship between two <a href="Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a>. Two variables are <i>perfectly collinear</i> if there is an exact linear relationship between the two, so the correlation between them is equal to 1 or −1. That is, <span class="texhtml"><i>X</i><sub>1</sub></span> and <span class="texhtml"><i>X</i><sub>2</sub></span> are perfectly collinear if there exist parameters <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}}">

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</math></span><img src="./cfa5ad1eb6cdaf3d8dfd77991ee9ce7bdf169184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \lambda _{0}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{2i}=\lambda _{0}+\lambda _{1}X_{1i}.}</annotation>

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</math></span><img src="./f12f9bef943db04a9a01ffa6d2ca7179eaf1ae1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.496ex; height:2.509ex;" alt="{\displaystyle X_{2i}=\lambda _{0}+\lambda _{1}X_{1i}.}" loading="lazy"></span></dd></dl>

<p>This means that if the various observations <span class="texhtml">(<i>X</i><sub>1<i>i</i></sub>, <i>X</i><sub>2<i>i</i></sub>)</span> are plotted in the <span class="texhtml">(<i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>)</span> plane, these points are collinear in the sense defined earlier in this article.

</p><p>Perfect <b>multicollinearity</b> refers to a situation in which <span class="texhtml"><i>k</i> (<i>k</i> ≥ 2)</span> explanatory variables in a <a href="Multiple_regression" class="mw-redirect" title="Multiple regression">multiple regression</a> model are perfectly linearly related, according to

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<annotation encoding="application/x-tex">{\displaystyle X_{ki}=\lambda _{0}+\lambda _{1}X_{1i}+\lambda _{2}X_{2i}+\dots +\lambda _{k-1}X_{(k-1),i}}</annotation>

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<p>for all observations <span class="texhtml mvar" style="font-style:italic;">i</span>. In practice, we rarely face perfect multicollinearity in a data set. More commonly, the issue of multicollinearity arises when there is a "strong linear relationship" among two or more independent variables, meaning that

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<annotation encoding="application/x-tex">{\displaystyle X_{ki}=\lambda _{0}+\lambda _{1}X_{1i}+\lambda _{2}X_{2i}+\dots +\lambda _{k-1}X_{(k-1),i}+\varepsilon _{i}}</annotation>

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</math></span><img src="./d26c4359ba5ca181a1b3ed68fd5e7ff6a7cb8198.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.77ex; height:3.009ex;" alt="{\displaystyle X_{ki}=\lambda _{0}+\lambda _{1}X_{1i}+\lambda _{2}X_{2i}+\dots +\lambda _{k-1}X_{(k-1),i}+\varepsilon _{i}}" loading="lazy"></span></dd></dl>

<p>where the variance of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}}">

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</p><p>The concept of <i>lateral collinearity</i> expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Usage_in_other_areas">Usage in other areas</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Antenna_arrays">Antenna arrays</h3></div>

<p>In <a href="Telecommunication" class="mw-redirect" title="Telecommunication">telecommunications</a>, a <a href="Collinear_antenna_array" title="Collinear antenna array">collinear (or co-linear) antenna array</a> is an <a href="Antenna_array" title="Antenna array">array</a> of <a href="Dipole_antenna" title="Dipole antenna">dipole antennas</a> mounted in such a manner that the corresponding elements of each <a href="Antenna_(radio)" title="Antenna (radio)">antenna</a> are parallel and aligned, that is they are located along a common line or axis.

</p>

<div class="mw-heading mw-heading3"><h3 id="Photography">Photography</h3></div>

<p>The <a href="Collinearity_equation" title="Collinearity equation">collinearity equations</a> are a set of two equations, used in <a href="Photogrammetry" title="Photogrammetry">photogrammetry</a> and <a href="Computer_stereo_vision" title="Computer stereo vision">computer stereo vision</a>, to relate <a href="Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> in an image (<a href="Sensor" title="Sensor">sensor</a>) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering the <a href="Projection_(mathematics)" title="Projection (mathematics)">central projection</a> of a point of the <a href="3D_modeling" title="3D modeling">object</a> through the <a href="Cardinal_point_(optics)#Principal_planes_and_points" title="Cardinal point (optics)">optical centre</a> of the <a href="Pinhole_camera" title="Pinhole camera">camera</a> to the image in the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their image points are all concurrent at the optical centre.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Concyclic_points" title="Concyclic points">Concyclic points</a></li>

<li><a href="Coplanarity" title="Coplanarity">Coplanarity</a></li>

<li><a href="Direction_(geometry)" title="Direction (geometry)">Direction (geometry)</a></li>

<li><a href="Incidence_(geometry)#Collinearity" title="Incidence (geometry)">Incidence (geometry)#Collinearity</a></li>

<li><a href="No-three-in-line_problem" title="No-three-in-line problem">No-three-in-line problem</a></li>

<li><a href="Pappus's_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's hexagon theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The concept applies in any geometry <a href="#CITEREFDembowski1968">Dembowski (1968</a>, pg. 26), but is often only defined within the discussion of a specific geometry <a href="#CITEREFCoxeter1969">Coxeter (1969</a>, pg. 178), <a href="#CITEREFBrannanEsplenGray1998">Brannan, Esplen & Gray (1998</a>, pg.106)</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/colinear">Colinear (Merriam-Webster dictionary)</a></span>

</li>

<li id="cite_note-Johnson-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Johnson_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Johnson_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Johnson, Roger A., <i>Advanced Euclidean Geometry</i>, Dover Publ., 2007 (orig. 1929).</span>

</li>

<li id="cite_note-AC-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-AC_4-0">^</a></b></span> <span class="reference-text"><a href="Nathan_Altshiller_Court" title="Nathan Altshiller Court">Altshiller Court, Nathan</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/collegegeometryi00alts"><i>College Geometry</i></a>, 2nd ed. Barnes & Noble, 1952 [1st ed. 1925].</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry", <i><a href="Mathematical_Gazette" class="mw-redirect" title="Mathematical Gazette">Mathematical Gazette</a></i> 83, November 1999, 472–477.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, <i>The IMO Compendium</i>, Springer, 2006, p. 15.</span>

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</style><cite id="CITEREFMyakishev2006" class="citation cs2">Myakishev, Alexei (2006), <a rel="nofollow" class="external text" href="http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf">"On Two Remarkable Lines Related to a Quadrilateral"</a> <span class="cs1-format">(PDF)</span>, <i>Forum Geometricorum</i>, <b>6</b>: <span class="nowrap">289–</span>295</cite>.</span>

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<li id="cite_note-Honsberger-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Honsberger_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFHonsberger1995" class="citation cs2"><a href="Ross_Honsberger" title="Ross Honsberger">Honsberger, Ross</a> (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6oduPgvOAhwC&pg=PA35">"4.2 Cyclic quadrilaterals"</a>, <i>Episodes in Nineteenth and Twentieth Century Euclidean Geometry</i>, New Mathematical Library, vol. 37, Cambridge University Press, pp. <span class="nowrap">35–</span>39, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-639-0</bdi></cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFBradley2011" class="citation cs2">Bradley, Christopher (2011), <a rel="nofollow" class="external text" href="http://people.bath.ac.uk/masgcs/Article141.pdf"><i>Three Centroids created by a Cyclic Quadrilateral</i></a> <span class="cs1-format">(PDF)</span></cite></span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFKockLynn2012" class="citation journal cs1">Kock, N.; Lynn, G. S. (2012). <a rel="nofollow" class="external text" href="http://www.scriptwarp.com/warppls/pubs/Kock_Lynn_2012.pdf">"Lateral collinearity and misleading results in variance-based SEM: An illustration and recommendations"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the Association for Information Systems</i>. <b>13</b> (7): <span class="nowrap">546–</span>580. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17705%2F1jais.00302">10.17705/1jais.00302</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3677154">3677154</a>.</cite></span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">It's more mathematically natural to refer to these equations as <i>concurrency equations</i>, but photogrammetry literature does not use that terminology.</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBrannanEsplenGray1998" class="citation cs2">Brannan, David A.; Esplen, Matthew F.; <a href="Jeremy_Gray_(mathematician)" title="Jeremy Gray (mathematician)">Gray, Jeremy J.</a> (1998), <i>Geometry</i>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-59787-0</bdi></cite></li>

<li><cite id="CITEREFCoxeter1969" class="citation cs2"><a href="H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">Coxeter, H. S. M.</a> (1969), <i>Introduction to Geometry</i>, New York: John Wiley & Sons, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-50458-0</bdi></cite></li>

<li><cite id="CITEREFDembowski1968" class="citation cs2"><a href="Peter_Dembowski" title="Peter Dembowski">Dembowski, Peter</a> (1968), <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/finitegeometries0000demb"><i>Finite geometries</i></a></span>, <a href="Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete" title="Ergebnisse der Mathematik und ihrer Grenzgebiete">Ergebnisse der Mathematik und ihrer Grenzgebiete</a>, vol. 44, Berlin: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-61786-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0233275">0233275</a></cite></li></ul></div><div><div class="zim-footer">

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